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Rescattering effects in heavy flavour production off nuclei
ELSEVIER
Nuclear Physics B (Proc. Suppl.) 71 (1999) 254-258
PROCEEDINGS SUPPLEMENTS
Rescattering effects in heavy flavour production off nuclei M.A. Braun *, C.Pajares and C.A. Salgado, a N.Armesto and A.Capella b aDepartamento de Fisica de Particulas, Universidad de Santiago de Compostela, 15706 Santiago de Compostela, Spain bLaboratoire de Physique Theorique et Hautes Energies, Universite de Paris XI, Batiment 211, F-91405 Orsay Cedex, France The absorptive corrections resulting from the rescattering of a heavy flavour (the so-called nuclear absorption) are usually calculated with a probabilistic formula valid only in the low energy limit. We extend this formula
to all energies using a quantum field theoretical approach. For charmonium and bottonium we find that the absorptive corrections in the rigorous treatment are very similar to the ones in the probabilistic approach. On the contrary, at sufficiently high energy, open charm and bottom are absorbed as much as charmonium and bottonium - in spite of the fact that their absorptive cross-sections are zero, and therefore they are not absorbed in the probabilistic model. At high enough energies there are also absorptive corrections due to the shadowing of the nucleus structure function, which are present for all systems including Drell-Yan pair production. These shadowing corrections cancel in the low energy limit.
1. I n t r o d u c t i o n
In heavy flavour production on nuclei rescattering of the pre-resonant QQ, system in the target is currently described by the probabilistic formula [1,2].
Ia-- nabs l f
IN
(1)
"¢2
Instead of successive interactions of the projectile with nucleons of the target, one has simultaneous interactions of particles into which the projectile has split. We are going to study the changes in Eq. (1) resulting from this change in the physical picture at high energies. We find an expression for the absorption factor valid at all energies, which is different from (1) but goes into it in the low energy limit. In this report we only present our results. Their detailed derivation may be found in [3].
Here TA is the transverse nuclear density; IA(]V) are the inclusive J/@ cross-section on A(N) and a~bs is the absorptive (Q(~) - N cross-section . For open heavy flavour production a abs = 0 and one obtains an A 1 behaviour, whereas for charmonium and bottonium a ab" ~ 0 and one obtains a behaviour A ~ with a < 1, in agreement with experiment. Eq. (1) has a probabilistic interpretation with a clear longitudinal ordering in z: in the first interaction at z the heavy system is produced and in successive ones at z' > z it rescatters with nucleons along its path. However, there is a fundamental change in the physical picture as one goes from low energies, for which the probabilistic formula (1) is derived, to high energies.
In the diagrammatic language production of a heavy particle is currently described by the diagram shown in Fig. 1. The corresponding inclusive cross-section is essentially a product of two gluonic densities in the projectile and target and a hard cross-section to produce the heavy particle by gluon fusion. For pN collisions, integrated over transverse momenta, it is given by
*Permanent address: Department of High Energy Physics, University of S.Petersburg, 198904 S.Petersburg, Russia
IN(X) = --~"~Fp(xl, M2)FN(x2, M2),
0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0920-5632(98)00351-X
2. D i a g r a m s c o n s i d e r e d tion made
7rg 2
and
approxima-
(2)
M.A. Braun et al./Nuclear Physics B (Proc. Suppl.) 71 (1999) 254-258
a
255
b
Figure 1. Production of a heavy flavoured particle in pN collisions. The heavy particle is indicated by a thick line.
Figure 2. Production of a heavy flavoured particle (thick line) in pA collisions via external (a) and internal (b) mechanisms.
Here gM corresponds to the hard scattering vertex, M is the heavy particle mass, xl and x2 are the longitudinal momentum fractions of the projectile and target carried by the colliding partons (gluons), x = Xl - x2 is the heavy particle Feynman scaling variable, M 2 = XlX2S where s is the c.m energy squared. Fp(g)(Xl,M 2) is the structure function of the projectile proton (nucleon) at Q2 = M 2. Passing to hA collisions we have to consider discontinuities in the total energy of the diagrams shown in Fig.2 a, b. Of these, Fig. 2 a is a clear generalization of Fig. 1 to a nucleus target. Note that it corresponds not only to the absorption of the produced particle due to its rescattering in the nucleus, which is described by the central blob, but also to the change of the gluon distribution in the nuclear matter (the EMC effect), described by the lower blob. This latter may be viewed upon as rescattering of the light parton (gluon) in the nucleus. The diagram Fig. 2 b refers to a somewhat different mechanism for the heavy particle production. It describes a process in which a heavy particle already present in the projectile (target) collides with a target (projectile) and is observed
afterwards. We call this contribution internal, to distinguish from the external contribution from Fig. 2 a. Note that our internal contribution does not coincide with the so-called intrinsic one, since both in the projectile and target the heavy flavour may be generated perturbatively. The intrinsic contribution corresponds to an unperturbative part of the intrinsic one. In the following we assume that it is absent. Apart from this approximation, to render treatment feasible, we make some more simplifications. Essentially they are three. 1) We treat the heavy flavoured system produced (e.g a charmed QQ pair) as a single particle of a single sort, which we call J/~ for convenience. 2) We neglect all spins. 3) We treat all interactions with the nucleus in the eikonal approximation. It is this last assumption that reduces our picture of nuclear absorption to a standard one, excluding all collective and non-linear effects, like the quark-gluon plasma formation or interaction with co-movers.
256
M.A. Braun et al./Nuclear Physics B (Proc. Suppl.) 71 (1999) 254-258
3. R e s u l t s for a s y m p t o t i c e n e r g i e s
Our results are most easily formulated for p-A interactions at very high (asymptotic) energies. For the external contribution (Fig. 2 a) we obtain a formula, which generalizes (1),
IA(e~t)= r(~t) "g 1 [aA(a + a,) - aA(aci].
(3)
r(ext) is given by (2); aA(a~) is the total Here ~N cross-section on the nucleus calculated with the - N forward scattering amplitude a~,. The amplitude a refers to the light patton-nucleon scattering and is, in fact, a parameter in our approach. Its value determines the EMC effect. If we neglect it, for the pure nuclear absorption we get
in the absorptive corrections due to the change in the space-time picture results comparatively small and the probabilistic expression remains approximately valid even at LHC energies (see the numerical results in Sec. 4). The situation is completely different for open charm or bottom production. In this case a ab8 = 0 and (1) leads to the cross-section ,~ A 1. Our results indicate, however, that with increasing energy there appear shadowing corrections due to nuclear rescattering, which at very high energies become of the same order as those for the J/¢~. This is a main prediction of our approach. These results are easily generalized to the nucleus-nucleus scattering. For the external part we find a factorization formula
iA(ext) B
IAext) I'* Ir(eXt) N
A f d2bTA(b)e ia*ATA(b)
(5)
The internal contribution from the projectile is, however, absorbed differently i A(in,P) l r(in,p)
,'N
where a(~) ..(a) ¢1A ---- V ~ A
= a~)A/(ea~) _(~n)
+ °~a
_(in) ~_
-- OkOA ~t~kO ~
"A
"B
(8)
RA (4)
The internal contribution from the target is absorbed exactly in the same manner ia(in,a) / r ( i n , N ) I-~N "= R A
r(ezt) t(ext)
that is,the total absorption factor is just the product of absorptive factors coming from both nuclei. As is well-known this relation also holds in the probabilistic approach [1,2]. For the internal parts we also obtain expressions factorized in the two colliding nuclei. The internal heavy flavour present in the nucleus B gives a contribution
(6)
r( int,B ) r(int,B) l(int,N) IB • aB = -A
a~(1 - e)) (7)
and e < 1 is a phenomenological parameter, introduced to account for possible transitions from hidden to open heavy flavour. For the open heavy flavour e = 1. To discuss these formulas, one has to take into account that, as numerical calculation show, for x > 0 the internal contribution from the projectile is neglegible. So the total absorptive factor is RA. Comparing to (1) we see that it is different from the probabilistic factor. For large A it behaves like a diffractive cross-section on the nucleus, rather than the inelastic one in (1). However for a¢ ~ a ~b~, which is the case for charmonium or bottonium production, the lowest order rescattering correction is the same. Since a . is small, the change with increasing energy
(9)
"N
To this internal contribution a similar one I(~ t'A) has to be added, which takes into account the internal heavy flavour of the target. It is given by (9) with the substitution A o B. 4. F i n i t e e n e r g y c o r r e c t i o n . N u m e r i c a l resuits
As discussed in [4,5], finite energy corrections have a clear origin. Due to the presence of the heavy system, some of the contributions to the inclusive cross-section have a non-vanishing minimal transverse momentum (train # 0) and are suppressed by the nuclear form factor in a welldefined way. These modified cutting rules have been found in [5] in the framework of a specific patton model. However, their physical content is
257
M.A. Braun et al./Nuclear Physics B (Proc. Suppl.) 71 (1999) 254-258
so transparent that they presumably have a more general validity. They can be summarized in the following way. Let us consider a particular ordering of the longitudinal coordinates zj of n interactions with the nucleus zl <_ z2 <_ . . . <_ Zn. Then at finite energies, the n-th power of the nucleus profile function T~ has to be replaced by one of the following integrals [5]: T (j) = n!
dzipA(b, zi)
e i~zlj
(10)
where z0 = - e c , Zlj = Zl - zj, PA is the nuclear density, j = 1,2,...n and A = m N M 2 / s x . For A = 0, corresponding to asymptotic energies, all integrals T (j) are equal to T~ and are independent
of j. For A non zero the value of j to be taken in Eq. (10) depends on a particular discontinuity of the scattering amplitude considered (a "cut diagram"). All the discontinuities containing T (j) are of the following type [5]. Interactions with the nucleus from 1 to j - 1 have to be located to the left of the cutting line. Interaction j may either be cut or be located to the right of the cutting line. All the other interactions, from j + 1 to n may be cut in all possible ways. To this contribution one has to add its complex conjugate. The physical content of these rules is quite clear. When the first interaction is cut the exponential damping factor in (10) is not present, i.e. T(1) = T~. Clearly this is the only case where train -- 0 [4]. In all other cases the exponential damping factor is present and depends on the longitudinal distance zj - zl between the interaction 1 and the first one j which is either cut or lies to the right of the cutting line. Using these rules it is straightforward to write the expression for the inclusive cross-section at finite energies (i.e. A ~ 0). Its explicit form is rather complicated and can be found in [3]. The important point is that from this expression it follows that al low energies (i.e. A --+ oc) all screening corrections to the gluon distribution in the nucleus (the EMC effect) vanish and that the total nuclear absorption is exactly given by the probabilistic formula (1). As far as we know this is the first time that this expression has been de-
rived in a field theoretical approach. Using the obtained formulas we have calculated charm production in pPb and Pb-Pb collisions at various energies. We have taken e = 0.001 and 0.999 for hidden and open charm, respectively. The q2 - p cross-section was taken to be a s = 7 mbn at s = 60 G e V 2 in accordance with the data [6]. For the gluon densities Fp,N we have taken the GRV LO parametrization [7]. The EMC effect has been taken into account by introducing a reduction factor due to nuclear corrections to the structure functions found in [8].
Table 1 Effective atomic numbers for J/g1 production in p-Pb collisions. X/~, G e V
Aeff
Aprob "'elf
XF=O
20 39 200 6000
147.2 137.5 102.4 71.0
134.4 131.7 124.7 109.5
XF=0.5
20 39 200 6000
121.0 97.3 70.5 43.1
129.1 123.6 109.1 78.6
Our results are presented in Tables 1-3 for the production of J/g2 and open charm in p-Pb collisions and of J/@ in Pb-Pb collisions respectively. For J / ~ also the results following from the probabilistic formula are presented for comparison. In this case one observes that in the central region (x = 0) the difference between our relativistic treatment and the probabilistic one is quite small up to c.m. energies of the order 200 G e V . On the whole the relativistic theory predicts more absorption, which may become twice as strong at high energy and x, as compared to the probabilistic treatment. One also notices that the resulting x-dependence turns out to be consistent with the experimental one [9].
M.A. Braun et al./Nuclear Physics 13 (Proc. Suppl.) 71 (1999) 254-258
258
Table 2 Effective atomic numbers for open charm production in p-Pb collisions.
v/s, GeV
Aeff
Aasym
"'ell
XF~O
20 39 200 6000
222.0 186.9 105.1 71.0
144.4 131.9 101.9 71.0
the probabilistic formula up to x/s ~- 6 TeV. For open heavy flavour production we predict nuclear absorption for v ~ ,-- 40 GeV, which turns out to be almost the same as the suppression of the J / q at higher energies. Our formalism predicts an increase of the J/@ suppression with increasing XF. Together with the corresponding increase of the shadowing corrections in the nuclear structure functions, it provides an explanation of the observed xF-dependence of J / q suppression.
XF=0.5 20 39 200 6000
139.5 98.7 70.3 42.7
116.8 96.8 70.3 43.1
Table 3 Effective atomic numbers for J / ~ production in Pb-Pb collisions atxF=O.
v ~, GeV
A~I!
Aprob
20 39 200 6000
18230 17830 14132 9943
18080 17340 15550 12000
"*~I!
For the open charm we have also presented results from the asymptotical formulae. As one sees, the finite energy effects are very strong. At small energies they make the absorption quite small (~ ,~ 0.98 in the A-dependence AS), bringing the results in accordance with the experimental data [10]. This effect only dies out at x/s _> 200 GeV, when our model predicts an absorption for the open charm of the same order as for the hidden one. 5. C o n c l u s i o n s The probabilistic formula used up to now to describe heavy flavour production off nuclei has been generalized to all energies using a quantum field theoretical approach. For J / ~ and T production it gives practically the same results as
6.
Acknowledgements
We are grateful to Yu. M. Shabelski for fruitful discussions. We thank the Direcci6n General de Polftica Cientffica and CICYT for financial support under contract AEN96-1673. C.A.S. also thanks Xunta de Galicia for a grant. REFERENCES 1. A. Capella, J. A. Casado, C. Pajares, A. V. Ramallo and J. Tran Thanh Van, Phys. Lett. B206 (1988) 354. 2. C. Gerschel and J. Huefner, Phys. Lett. B 2 0 7 (1988) 354. 3. M.A. Braun et al., preprint US-FT/22-97, DESY 97-149 (hep-ph/9707424), to be published in Nucl. Phys. B. 4. K. Boreskov, A. Capella, A. Kaidalov and J. Tran Thanh Van, Phys. Rev. D 4 7 (1993) 919. 5. M. A. Braun and A. Capella, Nucl. Phys. B412 (1994) 260.18 (1973) 595. 6. D. Kharzeev et al., Z. Phys. C74 (1997) 307; N. Armesto and A. Capella LPTHE Orsay 97/11 preprint (hep-ph/9705275). 7. M. Glfick, E. Reya and A. Vogt, Z. Phys. C67 (1995) 433. 8. K.J. Eskola Nucl. Phys B400 (1993) 240; N. Armesto, C. Pajares, C. A. Salgado and Yu. M. Shabelski, Phys. Lett. B366 (1996) 276 and US-FT/37-96 preprint (hep-ph/9609296) (to be published in Phys. At. Nucl.). 9. D.M. Alde et al., Phys. Rev. Lett. 66 (1991) 133. 10. G. A. Alves et al., Phys. Rev. Lett. 70 (1993) 722; E789 collaboration, M. J. Leitch et al., Phys. Rev. Lett. 72 (1994) 2542.
Nuclear Physics B (Proc. Suppl.) 71 (1999) 254-258
PROCEEDINGS SUPPLEMENTS
Rescattering effects in heavy flavour production off nuclei M.A. Braun *, C.Pajares and C.A. Salgado, a N.Armesto and A.Capella b aDepartamento de Fisica de Particulas, Universidad de Santiago de Compostela, 15706 Santiago de Compostela, Spain bLaboratoire de Physique Theorique et Hautes Energies, Universite de Paris XI, Batiment 211, F-91405 Orsay Cedex, France The absorptive corrections resulting from the rescattering of a heavy flavour (the so-called nuclear absorption) are usually calculated with a probabilistic formula valid only in the low energy limit. We extend this formula
to all energies using a quantum field theoretical approach. For charmonium and bottonium we find that the absorptive corrections in the rigorous treatment are very similar to the ones in the probabilistic approach. On the contrary, at sufficiently high energy, open charm and bottom are absorbed as much as charmonium and bottonium - in spite of the fact that their absorptive cross-sections are zero, and therefore they are not absorbed in the probabilistic model. At high enough energies there are also absorptive corrections due to the shadowing of the nucleus structure function, which are present for all systems including Drell-Yan pair production. These shadowing corrections cancel in the low energy limit.
1. I n t r o d u c t i o n
In heavy flavour production on nuclei rescattering of the pre-resonant QQ, system in the target is currently described by the probabilistic formula [1,2].
Ia-- nabs l f
IN
(1)
"¢2
Instead of successive interactions of the projectile with nucleons of the target, one has simultaneous interactions of particles into which the projectile has split. We are going to study the changes in Eq. (1) resulting from this change in the physical picture at high energies. We find an expression for the absorption factor valid at all energies, which is different from (1) but goes into it in the low energy limit. In this report we only present our results. Their detailed derivation may be found in [3].
Here TA is the transverse nuclear density; IA(]V) are the inclusive J/@ cross-section on A(N) and a~bs is the absorptive (Q(~) - N cross-section . For open heavy flavour production a abs = 0 and one obtains an A 1 behaviour, whereas for charmonium and bottonium a ab" ~ 0 and one obtains a behaviour A ~ with a < 1, in agreement with experiment. Eq. (1) has a probabilistic interpretation with a clear longitudinal ordering in z: in the first interaction at z the heavy system is produced and in successive ones at z' > z it rescatters with nucleons along its path. However, there is a fundamental change in the physical picture as one goes from low energies, for which the probabilistic formula (1) is derived, to high energies.
In the diagrammatic language production of a heavy particle is currently described by the diagram shown in Fig. 1. The corresponding inclusive cross-section is essentially a product of two gluonic densities in the projectile and target and a hard cross-section to produce the heavy particle by gluon fusion. For pN collisions, integrated over transverse momenta, it is given by
*Permanent address: Department of High Energy Physics, University of S.Petersburg, 198904 S.Petersburg, Russia
IN(X) = --~"~Fp(xl, M2)FN(x2, M2),
0920-5632/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0920-5632(98)00351-X
2. D i a g r a m s c o n s i d e r e d tion made
7rg 2
and
approxima-
(2)
M.A. Braun et al./Nuclear Physics B (Proc. Suppl.) 71 (1999) 254-258
a
255
b
Figure 1. Production of a heavy flavoured particle in pN collisions. The heavy particle is indicated by a thick line.
Figure 2. Production of a heavy flavoured particle (thick line) in pA collisions via external (a) and internal (b) mechanisms.
Here gM corresponds to the hard scattering vertex, M is the heavy particle mass, xl and x2 are the longitudinal momentum fractions of the projectile and target carried by the colliding partons (gluons), x = Xl - x2 is the heavy particle Feynman scaling variable, M 2 = XlX2S where s is the c.m energy squared. Fp(g)(Xl,M 2) is the structure function of the projectile proton (nucleon) at Q2 = M 2. Passing to hA collisions we have to consider discontinuities in the total energy of the diagrams shown in Fig.2 a, b. Of these, Fig. 2 a is a clear generalization of Fig. 1 to a nucleus target. Note that it corresponds not only to the absorption of the produced particle due to its rescattering in the nucleus, which is described by the central blob, but also to the change of the gluon distribution in the nuclear matter (the EMC effect), described by the lower blob. This latter may be viewed upon as rescattering of the light parton (gluon) in the nucleus. The diagram Fig. 2 b refers to a somewhat different mechanism for the heavy particle production. It describes a process in which a heavy particle already present in the projectile (target) collides with a target (projectile) and is observed
afterwards. We call this contribution internal, to distinguish from the external contribution from Fig. 2 a. Note that our internal contribution does not coincide with the so-called intrinsic one, since both in the projectile and target the heavy flavour may be generated perturbatively. The intrinsic contribution corresponds to an unperturbative part of the intrinsic one. In the following we assume that it is absent. Apart from this approximation, to render treatment feasible, we make some more simplifications. Essentially they are three. 1) We treat the heavy flavoured system produced (e.g a charmed QQ pair) as a single particle of a single sort, which we call J/~ for convenience. 2) We neglect all spins. 3) We treat all interactions with the nucleus in the eikonal approximation. It is this last assumption that reduces our picture of nuclear absorption to a standard one, excluding all collective and non-linear effects, like the quark-gluon plasma formation or interaction with co-movers.
256
M.A. Braun et al./Nuclear Physics B (Proc. Suppl.) 71 (1999) 254-258
3. R e s u l t s for a s y m p t o t i c e n e r g i e s
Our results are most easily formulated for p-A interactions at very high (asymptotic) energies. For the external contribution (Fig. 2 a) we obtain a formula, which generalizes (1),
IA(e~t)= r(~t) "g 1 [aA(a + a,) - aA(aci].
(3)
r(ext) is given by (2); aA(a~) is the total Here ~N cross-section on the nucleus calculated with the - N forward scattering amplitude a~,. The amplitude a refers to the light patton-nucleon scattering and is, in fact, a parameter in our approach. Its value determines the EMC effect. If we neglect it, for the pure nuclear absorption we get
in the absorptive corrections due to the change in the space-time picture results comparatively small and the probabilistic expression remains approximately valid even at LHC energies (see the numerical results in Sec. 4). The situation is completely different for open charm or bottom production. In this case a ab8 = 0 and (1) leads to the cross-section ,~ A 1. Our results indicate, however, that with increasing energy there appear shadowing corrections due to nuclear rescattering, which at very high energies become of the same order as those for the J/¢~. This is a main prediction of our approach. These results are easily generalized to the nucleus-nucleus scattering. For the external part we find a factorization formula
iA(ext) B
IAext) I'* Ir(eXt) N
A f d2bTA(b)e ia*ATA(b)
(5)
The internal contribution from the projectile is, however, absorbed differently i A(in,P) l r(in,p)
,'N
where a(~) ..(a) ¢1A ---- V ~ A
= a~)A/(ea~) _(~n)
+ °~a
_(in) ~_
-- OkOA ~t~kO ~
"A
"B
(8)
RA (4)
The internal contribution from the target is absorbed exactly in the same manner ia(in,a) / r ( i n , N ) I-~N "= R A
r(ezt) t(ext)
that is,the total absorption factor is just the product of absorptive factors coming from both nuclei. As is well-known this relation also holds in the probabilistic approach [1,2]. For the internal parts we also obtain expressions factorized in the two colliding nuclei. The internal heavy flavour present in the nucleus B gives a contribution
(6)
r( int,B ) r(int,B) l(int,N) IB • aB = -A
a~(1 - e)) (7)
and e < 1 is a phenomenological parameter, introduced to account for possible transitions from hidden to open heavy flavour. For the open heavy flavour e = 1. To discuss these formulas, one has to take into account that, as numerical calculation show, for x > 0 the internal contribution from the projectile is neglegible. So the total absorptive factor is RA. Comparing to (1) we see that it is different from the probabilistic factor. For large A it behaves like a diffractive cross-section on the nucleus, rather than the inelastic one in (1). However for a¢ ~ a ~b~, which is the case for charmonium or bottonium production, the lowest order rescattering correction is the same. Since a . is small, the change with increasing energy
(9)
"N
To this internal contribution a similar one I(~ t'A) has to be added, which takes into account the internal heavy flavour of the target. It is given by (9) with the substitution A o B. 4. F i n i t e e n e r g y c o r r e c t i o n . N u m e r i c a l resuits
As discussed in [4,5], finite energy corrections have a clear origin. Due to the presence of the heavy system, some of the contributions to the inclusive cross-section have a non-vanishing minimal transverse momentum (train # 0) and are suppressed by the nuclear form factor in a welldefined way. These modified cutting rules have been found in [5] in the framework of a specific patton model. However, their physical content is
257
M.A. Braun et al./Nuclear Physics B (Proc. Suppl.) 71 (1999) 254-258
so transparent that they presumably have a more general validity. They can be summarized in the following way. Let us consider a particular ordering of the longitudinal coordinates zj of n interactions with the nucleus zl <_ z2 <_ . . . <_ Zn. Then at finite energies, the n-th power of the nucleus profile function T~ has to be replaced by one of the following integrals [5]: T (j) = n!
dzipA(b, zi)
e i~zlj
(10)
where z0 = - e c , Zlj = Zl - zj, PA is the nuclear density, j = 1,2,...n and A = m N M 2 / s x . For A = 0, corresponding to asymptotic energies, all integrals T (j) are equal to T~ and are independent
of j. For A non zero the value of j to be taken in Eq. (10) depends on a particular discontinuity of the scattering amplitude considered (a "cut diagram"). All the discontinuities containing T (j) are of the following type [5]. Interactions with the nucleus from 1 to j - 1 have to be located to the left of the cutting line. Interaction j may either be cut or be located to the right of the cutting line. All the other interactions, from j + 1 to n may be cut in all possible ways. To this contribution one has to add its complex conjugate. The physical content of these rules is quite clear. When the first interaction is cut the exponential damping factor in (10) is not present, i.e. T(1) = T~. Clearly this is the only case where train -- 0 [4]. In all other cases the exponential damping factor is present and depends on the longitudinal distance zj - zl between the interaction 1 and the first one j which is either cut or lies to the right of the cutting line. Using these rules it is straightforward to write the expression for the inclusive cross-section at finite energies (i.e. A ~ 0). Its explicit form is rather complicated and can be found in [3]. The important point is that from this expression it follows that al low energies (i.e. A --+ oc) all screening corrections to the gluon distribution in the nucleus (the EMC effect) vanish and that the total nuclear absorption is exactly given by the probabilistic formula (1). As far as we know this is the first time that this expression has been de-
rived in a field theoretical approach. Using the obtained formulas we have calculated charm production in pPb and Pb-Pb collisions at various energies. We have taken e = 0.001 and 0.999 for hidden and open charm, respectively. The q2 - p cross-section was taken to be a s = 7 mbn at s = 60 G e V 2 in accordance with the data [6]. For the gluon densities Fp,N we have taken the GRV LO parametrization [7]. The EMC effect has been taken into account by introducing a reduction factor due to nuclear corrections to the structure functions found in [8].
Table 1 Effective atomic numbers for J/g1 production in p-Pb collisions. X/~, G e V
Aeff
Aprob "'elf
XF=O
20 39 200 6000
147.2 137.5 102.4 71.0
134.4 131.7 124.7 109.5
XF=0.5
20 39 200 6000
121.0 97.3 70.5 43.1
129.1 123.6 109.1 78.6
Our results are presented in Tables 1-3 for the production of J/g2 and open charm in p-Pb collisions and of J/@ in Pb-Pb collisions respectively. For J / ~ also the results following from the probabilistic formula are presented for comparison. In this case one observes that in the central region (x = 0) the difference between our relativistic treatment and the probabilistic one is quite small up to c.m. energies of the order 200 G e V . On the whole the relativistic theory predicts more absorption, which may become twice as strong at high energy and x, as compared to the probabilistic treatment. One also notices that the resulting x-dependence turns out to be consistent with the experimental one [9].
M.A. Braun et al./Nuclear Physics 13 (Proc. Suppl.) 71 (1999) 254-258
258
Table 2 Effective atomic numbers for open charm production in p-Pb collisions.
v/s, GeV
Aeff
Aasym
"'ell
XF~O
20 39 200 6000
222.0 186.9 105.1 71.0
144.4 131.9 101.9 71.0
the probabilistic formula up to x/s ~- 6 TeV. For open heavy flavour production we predict nuclear absorption for v ~ ,-- 40 GeV, which turns out to be almost the same as the suppression of the J / q at higher energies. Our formalism predicts an increase of the J/@ suppression with increasing XF. Together with the corresponding increase of the shadowing corrections in the nuclear structure functions, it provides an explanation of the observed xF-dependence of J / q suppression.
XF=0.5 20 39 200 6000
139.5 98.7 70.3 42.7
116.8 96.8 70.3 43.1
Table 3 Effective atomic numbers for J / ~ production in Pb-Pb collisions atxF=O.
v ~, GeV
A~I!
Aprob
20 39 200 6000
18230 17830 14132 9943
18080 17340 15550 12000
"*~I!
For the open charm we have also presented results from the asymptotical formulae. As one sees, the finite energy effects are very strong. At small energies they make the absorption quite small (~ ,~ 0.98 in the A-dependence AS), bringing the results in accordance with the experimental data [10]. This effect only dies out at x/s _> 200 GeV, when our model predicts an absorption for the open charm of the same order as for the hidden one. 5. C o n c l u s i o n s The probabilistic formula used up to now to describe heavy flavour production off nuclei has been generalized to all energies using a quantum field theoretical approach. For J / ~ and T production it gives practically the same results as
6.
Acknowledgements
We are grateful to Yu. M. Shabelski for fruitful discussions. We thank the Direcci6n General de Polftica Cientffica and CICYT for financial support under contract AEN96-1673. C.A.S. also thanks Xunta de Galicia for a grant. REFERENCES 1. A. Capella, J. A. Casado, C. Pajares, A. V. Ramallo and J. Tran Thanh Van, Phys. Lett. B206 (1988) 354. 2. C. Gerschel and J. Huefner, Phys. Lett. B 2 0 7 (1988) 354. 3. M.A. Braun et al., preprint US-FT/22-97, DESY 97-149 (hep-ph/9707424), to be published in Nucl. Phys. B. 4. K. Boreskov, A. Capella, A. Kaidalov and J. Tran Thanh Van, Phys. Rev. D 4 7 (1993) 919. 5. M. A. Braun and A. Capella, Nucl. Phys. B412 (1994) 260.18 (1973) 595. 6. D. Kharzeev et al., Z. Phys. C74 (1997) 307; N. Armesto and A. Capella LPTHE Orsay 97/11 preprint (hep-ph/9705275). 7. M. Glfick, E. Reya and A. Vogt, Z. Phys. C67 (1995) 433. 8. K.J. Eskola Nucl. Phys B400 (1993) 240; N. Armesto, C. Pajares, C. A. Salgado and Yu. M. Shabelski, Phys. Lett. B366 (1996) 276 and US-FT/37-96 preprint (hep-ph/9609296) (to be published in Phys. At. Nucl.). 9. D.M. Alde et al., Phys. Rev. Lett. 66 (1991) 133. 10. G. A. Alves et al., Phys. Rev. Lett. 70 (1993) 722; E789 collaboration, M. J. Leitch et al., Phys. Rev. Lett. 72 (1994) 2542.